Question

analyze the graphs of f(x) and g(x). write an equation for g(x) in terms of f(x).
7 g(x) =

Explanation:

Step1: Analyze Key Points

For \( f(x) \), points are \((0,0)\), \((1,1)\), \((2,8)\). For \( g(x) \), points are \((0,2)\), \((1,1)\), \((2,-6)\). Notice \((1,1)\) is common. At \( x = 0 \), \( f(0)=0 \), \( g(0)=2 \); at \( x = 2 \), \( f(2)=8 \), \( g(2)=-6 \). Let's check transformations.

Step2: Check Reflection and Vertical Shift

First, check if there's a reflection over the x - axis. For \( x = 2 \), \( -f(2)=-8 \), but \( g(2)=-6 \), not -8. Wait, maybe horizontal reflection? Wait, \( g(x) \) at \( x = 0 \) is 2, \( f(0)=0 \). Let's check \( g(x)=-f(x)+2 \)? Wait, at \( x = 1 \), \( -f(1)+2=-1 + 2 = 1 \), which matches. At \( x = 2 \), \( -f(2)+2=-8 + 2=-6 \), which matches. At \( x = 0 \), \( -f(0)+2=0 + 2 = 2 \), which matches. So the transformation is a reflection over the x - axis (multiply by -1) and a vertical shift up by 2 units. So \( g(x)=-f(x)+2 \) or \( g(x)=2 - f(x) \).

Answer:

\( g(x)=-f(x)+2 \) (or equivalent form like \( g(x)=2 - f(x) \))